Adsorption of Ethylene on Neutral, Anionic and Cationic Gold Clusters

TL;DR

This study uses density-functional theory to analyze how ethylene adsorbs on small neutral, anionic, and cationic gold clusters, revealing different bonding modes, size-dependent energies, and charge effects that influence adsorption behavior.

Contribution

It provides a detailed theoretical comparison of ethylene adsorption modes on various charged gold clusters, highlighting the role of electronic shell effects and charge in adsorption mechanisms.

Findings

Pi-bonded adsorption dominates over di-sigma mode in most cases.

Adsorption energy varies with cluster size and charge state.

Vibrational frequency shifts can distinguish adsorption modes.

Abstract

The adsorption of ethylene molecule on neutral, anionic and cationic gold clusters consisting of up to 10 atoms has been investigated using density-functional theory. It is demonstrated that C2H4 can be adsorbed on small gold clusters in two different configurations, corresponding to the pi- and di-sigma-bonded species. Adsorption in the pi-bonded mode dominates over the di-sigma mode over all considered cluster sizes n, with the exception of the neutral C2H4-Au5 system. A striking difference is found in the size-dependence of the adsorption energy of C2H4 bonded to the neutral gold clusters in the pi and di-sigma configurations. The important role of the electronic shell effects in the di-sigma mode of ethylene adsorption on neutral gold clusters is demonstrated. It is shown that the interaction of C2H4 with small gold clusters strongly depends on their charge. The typical shift in the vibrational frequencies of C2H4 adsorbed in the pi- and the di-sigma configurations gives a guidance to experimentally distinguish between the two modes of adsorption.

Figures

Tables

Table 1. Table 1: Fundamental frequencies of C 2 ​ H 4 superscript subscript C 2 superscript subscript H 4 {{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}

Mode	Harmonic, cm-1	Experiment,72 cm-1
${\nu }_1$ $\left(b_{2u}\right)$	826	826
${\nu }_2$ $\left(b_{3u}\right)$	975	943
${\nu }_3$ $\left(b_{2g}\right)$	981	949
${\nu }_4$ $\left(a_u\right)$	1057	1023
${\nu }_5$ $\left(b_{3g}\right)$	1237	1236
${\nu }_6$ $\left(a_g\right)$	1377	1342
${\nu }_7$ $\left(b_{1u}\right)$	1467	1444
${\nu }_8$ $\left(a_g\right)$	1693	1623
${\nu }_9$ $\left(b_{1u}\right)$	3131	2989
${\nu }_{10}$ $\left(a_g\right)$	3146	3026
${\nu }_{11}$ $\left(b_{3g}\right)$	3205	3103
${\nu }_{12}$ $\left(b_{2u}\right)$	3233	3106

Equations

$$E_{\mathrm{ads}}=E_{tot}({\mathrm{Au}}_n)+E_{tot}({\mathrm{C}}_2{\mathrm{H}}_4)-E_{tot}({\mathrm{Au}}_n-{\mathrm{C}}_2{\mathrm{H}}_4),$$

Full text

\altaffiliation

On leave from: Institute of Physics, St Petersburg State University, 198504 St Petersburg, Petrodvorez, Russia

Hokkaido University]Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

1 Introduction

The adsorption of unsaturated hydrocarbons on transition metal surfaces has been studied extensively in order to understand the nature of hydrocarbon – metal interaction and chemical processes on solid surfaces; see, e.g., Refs. 1, 2, 3, 4, 5, 6 and references therein. The most significant attention was paid to investigation of ethylene adsorption, 7, 11, 12, 13, 14, 15, 10, 16, 8, 17, 9 because it is the simplest alkene containing an isolated carbon–carbon double bond. Hence it can be treated as a prototype to study the interaction and reactivity of different alkenes on metal surfaces. Moreover, the ethylene epoxidation is one of the most important processes in the chemical industry, because the product of such a reaction – ethylene oxide is widely used in various applications; see, e.g., Refs. 18, 19, 20, 21, 1, 2 and references therein.

Ethylene can adsorb on metal surfaces in two different configurations.11, 12, 14, 1 The first one is the $\pi$ mode where one metal atom on the surface is involved in the adsorption of ethylene via a $\pi$-bonding. The second one is the di-$\sigma$-bonded mode, when two metal atoms are involved in the adsorption via a $\sigma$ bonding. It was found that the di-$\sigma$ mode of adsorption is characterized by the increasingly important role of $sp^{3}$ hybridization in ethylene, while $sp^{2}$ hybridization (typical for free molecules) remains unchanged for $\pi$-bonded species.11, 12, 14

The bonding of ethylene with transition metals involves electron transfer from the filled bonding $\pi$ orbital of the ethylene to the metal, alongside a back-donation from the d-orbital of transition metal to the empty $\pi^{*}$ anti-bonding orbital of ethylene in accordance with the Dewar-Chatt-Duncanson model 22. Thus the appearance of the $sp^{3}$ character of hybridization in the di-$\sigma$-bonded ethylene can be explained by the increasing role of the electron back-donation from the metal to the $\pi^{*}$ anti-bonding orbital of ethylene. Hence the di-$\sigma$-bonded ethylene is activated more strongly in comparison with the $\pi$-bonded one, thus it can be more reactive.11, 12 Therefore, an understanding of the specific mechanisms of ethylene adsorption on metal surfaces is important in order to gain a better insight of the catalytic processes and reactivity of adsorbed hydrocarbons.

Surprisingly, not many works have been devoted to the investigation of ethylene adsorption on metal clusters and nanoparticles, despite the fact that chemical and physical properties of matter at nanoscale are very different from those of the corresponding bulk solids. These properties are often controlled by quantum size effects and significantly depend on the size and structure of atomic clusters.23

A remarkable example is gold. It is well known that gold in its bulk form is a catalytically inactive and inert metal. However gold at nanoscale manifests extraordinary catalytic activity which increases with a decrease in the cluster size of up to 1-5 nm 24, 25, 26, 27. Moreover, recent studies demonstrate that catalytic activity of gold clusters adsorbed on an iron oxide support correlates with the presence of very small clusters of $\sim$ 10 atoms 28. It was also reported that gold clusters with the number of atoms $n=3-11$ possess extraordinarily high electrocatalytic activity toward the ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ reduction reaction in acid solutions.29 The origin of such size-dependent catalytic activity of gold remains highly debated and has yet to be fully understood. A comprehensive survey of the field can be found in review papers and books; see, e.g., Refs. 23, 30, 31, 32, 33, 34. It was demonstrated that the catalytic activity of gold clusters depends on the type of the support material, the presence of defects in the material (e.g. F-center defects), the environment or special additives; see, e.g., Refs. 25, 35, 36, 31, 37 and references therein. However, experiments indicate that gold clusters deposited on an inert support can efficiently catalyze the process of styrene oxidation by dioxygen 27. Hence, the origin of catalytic activity derives from the cluster itself and therefore even free clusters can be effective catalysts.

Currently, a large variety of catalytic reactions on gold clusters has been studied experimentally. This includes the processes of catalytic oxidation of carbon monoxide, as well as more complex reactions such as alcohol oxidation, the direct synthesis of hydrogen peroxide and alkene epoxidation.27, 38, 26, 39

However, theoretical studies of the gold nanocatalysis have mainly focused on the investigation of adsorption and catalytic reaction of ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and CO.31, 36 Nonetheless, the theoretical study of the interaction of alkenes with gold clusters has been relatively unexplored with the exception of works on the binding of propene on gold and mixed gold-silver clusters40, 41, propene epoxidation42 and our recent work on cooperative adsorption of oxygen and ethylene on small gold clusters43.

In this paper we report the results of a systematic theoretical study of the adsorption of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on free neutral, anionic and cationic gold clusters consisting of up to 10 atoms. We find that ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbs on small gold clusters in two different configurations, corresponding to the $\pi$- and di-$\sigma$-bonded species. Adsorption in the $\pi$-bonded mode dominates over di-$\sigma$ mode over all considered cluster sizes $n$ with the exception of the neutral ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{5}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ system, where the di-$\sigma$-bonded configuration is energetically more favorable. We find a striking difference in the size dependence of the adsorption energy of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ bonded to neutral gold clusters in the $\pi$ and di-$\sigma$ configurations. Thus, the adsorption energy of the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ develops non-monotonically as a function of cluster size with a local minima at $n$=6. The adsorption energy, calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ exhibits pronounced odd-even oscillations, showing the importance of the electronic shell effects in the di-$\sigma$ mode of ethylene adsorption on gold clusters.

We also demonstrate that the interaction of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ with small gold clusters strongly depends on their charge. The excess of a positive or a negative charge on the gold cluster can change the balance between electron donation and back-donation processes. Hence ethylene adsorption and reactivity can be manipulated by the cluster charge. This effect can be particularly important for understanding the mechanisms of catalytic activation of ethylene adsorbed on small gold clusters.

2 Theoretical Methods

The calculations are carried out using density-functional theory (DFT). The hybrid Becke-type three-parameter exchange density functional paired with the gradient-corrected Perdew-Wang 91 correlation functional (B3PW91) 44 is used. The choice of the B3PW91 functional is consistent with our previous work on cooperative adsorption and catalytic oxidation of ethylene on small gold clusters.43 The obtained theoretical data on dissociation energies and bonding in ${{\text{Au}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (1.904 eV, 2.546 Å), and ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (7.596 eV, 1.324 Å) are in a good agreement with those of earlier experimental studies, ${{\text{Au}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (2.30 eV, 2.472 Å) 45, and ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (7.76 eV, 1.339 Å) 46. The standard LANL2DZ basis set of primitive Gaussians is used to expand the gold cluster orbitals formed by the $5s^{2}5p^{6}5d^{10}6s^{1}$ outer electrons of Au (19 electrons per atom). The remaining 60 core electrons of the Au atom are represented by the Hay-Wadt effective core potential accounting for relativistic effects 47. For carbon and hydrogen, the aug-cc-pVTZ basis set48 is employed. Calculations have been carried out with the use of the Gaussian 03 code 49.

The structural properties of neutral, anionic and cationic gold clusters have been the subject of numerous theoretical investigations; see, e.g., Refs. 30, 31, 32 for a review. In the present work the cluster geometries have been determined with the use of the cluster fusion algorithm50 which belongs to the class of genetic global optimization methods 51, 52. We have successfully used a similar approach to find the optimized geometries of various types of atomic clusters 53, 54, 55. The optimized structures of the gold clusters ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}, and ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ ($n=1-10$) obtained in the present work are presented in 1. These structures are in a good agreement with those reported in previous theoretical studies; see, e.g., Refs. 56, 57, 58, 59, 32.

In order to obtain the most stable configuration of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on neutral, anionic and cationic gold clusters, we have created a large number of starting geometries by adding ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ molecule in different positions (up to 30) on the surface of the most stable cluster and up to eight isomer structures of the corresponding ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}, and ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters. The starting structures have been optimized further without any geometry constraints.

3 Theoretical results

3.1 Geometry optimization for ethylene adsorbed on neutral, cationic and anionic

gold clusters

The results of the cluster geometry optimization for neutral ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, anionic {{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} and cationic ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters within the size range $n\leq 10$ are shown in 2, 3 and 4, respectively.

Gold clusters with adsorbed ethylene possess various isomer forms whose number grows dramatically with cluster size. Among the huge variety of different isomers, one can distinguish two specific adsorption modes that correspond to the $\pi$- and di-$\sigma$ configurations of the adsorbate. The appearance of two different modes of ethylene adsorption on small gold clusters is similar to that reported for ethylene adsorption on Au(100) and Au(111) surfaces14. However in the case of adsorption on the bulk, there are only two possible geometrical configurations of ethylene (we do not consider here $\mu$-bridging species14, because ethylene adsorption in such a configuration is not possible in the case of small clusters). Moreover, there is no structural changes of the bulk surface due to adsorption of ethylene. In the case of ethylene adsorption on small gold clusters, each of the $\pi$- and di-$\sigma$ configurations possess a variety of structural isomers. These isomers correspond to the adsorption in nonequivalent positions on the surface of clusters. In 2, 3 and 4 we present the most stable structures of $\pi$- and di-$\sigma$-bonded ethylene adsorbed on small gold clusters. The C – C and C – Au interatomic distances are given in angstroms.

The interaction of gold clusters with ethylene results in an electron donation from the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ to the gold cluster alongside a back-donation from the cluster to the adsorbate in accordance with the Dewar-Chatt-Duncanson model 22. The rearrangement of the electronic structure and electron transfers upon adsorption lead to a dramatic change in cluster geometry. The interplay of electronic and geometry shell effects, also influence the cluster structure 55.

Free neutral gold clusters possess planar, two-dimensional (2D) structures when the cluster size is up to 13 Au atoms.60, 59 In 1, we present the most stable structures obtained for Aun, ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ and {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} clusters ($n$ = 1, …, 10) within the B3PW91/LANL2DZ DFT method. 2 demonstrates that adsorption of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ in the $\pi$ configuration leads to a prevalence of three-dimensional (3D) structures of gold clusters for $n\geq 9$; the structures of small ${{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ${{\text{Au}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters transform to that which are energetically favorable for free cations ${{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ and ${{\text{Au}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$, respectively. For $n<9$, ethylene favorably adsorbs at the cluster edge where the carbon-carbon bond lay in the plane of the cluster with the exception of the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{6}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ system, where the carbon-carbon bond is located perpendicular to the cluster plane (see 2).

Adsorption of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ in the di-$\sigma$ configuration results in a considerable rearrangement in the structure of neutral gold clusters. Thus, the geometry of ${{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, and ${{\text{Au}_{\vphantom{\text{}}\text{6}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ – ${{\text{Au}_{\vphantom{\text{}}\text{10}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters changes upon ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption. It is known that gold clusters possess a large number of energetically close-lying structural isomers.30, 32 Interaction with adsorbate can easily alter their energy order. It is important to note, that interaction with adsorbate can be responsible for structural rearrangements even for the relatively large clusters. Thus, recent DFT calculations demonstrate that the morphology of ${{\text{Au}_{\vphantom{\text{}}\text{79}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ nanoparticle can be transformed by interactions with a CO atmosphere.61

It is not surprising that the structures of positively and negatively charged ionic gold clusters change upon ethylene adsorption. 3 demonstrates that although anionic gold clusters {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} with $n$ = 1, …, 10 remain planar upon ethylene adsorption (both for the $\pi$ and di-$\sigma$ adsorption configurations), they manifest considerable structural rearrangements. Thus, gold cluster anions alter their structure upon ethylene adsorption in the $\pi$-configuration for $n$ = 4, 7, 8, and 10. In the case of di-$\sigma$ mode of ethylene adsorption, structural rearrangements occur for gold cluster anions with $n$ = 3 – 8. A carbon-carbon bond of the adsorbed ethylene lie in a cluster plane for all structures presented in 3.

4 shows that in the case of ethylene adsorption on gold cluster cations, ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$, the resulting cluster systems ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ exhibit 3D structures for $n$= 5 – 10 in the case of the $\pi$ mode of ethylene adsorption; and $n$ = 8 and 10 in the case of the di-$\sigma$ mode. In both cases of the $\pi$ and di-$\sigma$ modes of ethylene adsorption, the structural rearrangements in gold cluster cations occur for $n\geq$ 5.

Such a severe change in the morphology of gold clusters due to the interaction with ethylene makes the procedure of structural optimization rather complicated. In order to get energetically favorable structures of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, {{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}, and ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters, it is necessary to consider adsorption of ethylene not only on the most stable structures of the corresponding ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}, and ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters, but also on a large number of their energetically less favorable isomer states. In the present work we have taken into account up to 8 isomer structures of the host gold clusters. This insures that we do not miss energetically favorable structures with considerable change in cluster geometry due to ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption.

3.2 Energetics of ethylene adsorption on neutral, anionic and cationic gold clusters

We first study the energetics of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption on neutral gold clusters Aun with $1\leq n\leq 10$. The spin states of the optimized ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ structures are doublet and singlet for odd and even $n$, respectively. 5 shows the evolution of the molecular adsorption energy, $E_{\rm ads}$, calculated for the most bounded $\pi$ and di-$\sigma$ configurations of adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ as a function of cluster size $n$. The adsorption energy is defined as

[TABLE]

where $E_{tot}(M)$ denotes the total energy of the most stable structure of the molecule (cluster) "$M$".

In our previous work, we have studied adsorption of ethylene on small neutral gold clusters in the $\pi$-bonded configuration.43 Here, we demonstrate that ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ can be adsorbed on small gold clusters in two different configurations, corresponding to the $\pi$- and di-$\sigma$-bonded species. 5 shows that the $\pi$-bonded mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption is energetically favorable in the considered range of cluster sizes $n$, with the exception of a single Au atom which interacts with one carbon atom in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, forming a $\sigma$ bond, and the ${{\text{Au}_{\vphantom{\text{}}\text{5}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ cluster, where the di-$\sigma$-bonded configuration of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is 0.05 eV more bounded compared with the $\pi$-bonded one.

It is seen from 5 that the adsorption energy of the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters exhibit a maximum at $n=$ 3 and 4 followed by a fast drop with a minimum at $n=$ 6. For $6\leq n\leq 10$ the adsorption energy of $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ shows a weak oscillatory behavior as a function of $n$.

The $n$-evolution of the adsorption energy calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is considerably different from that of the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. 5 demonstrates that the adsorption energy of the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ has an odd-even oscillatory behavior with a local maxima for odd numbers of Au atoms $n=$ 3, 5, 7, and 9. For clusters with an even number of $n$, the adsorption energy $E_{\rm ads}$ of the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is small. Hence, ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ readily adsorbs in the di-$\sigma$ configuration—but only for neutral gold clusters with an odd number of Au atoms.

It is interesting to note that in the case of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption on a Au(111) surface, the di-$\sigma$ mode ($E_{\rm ads}^{di-\sigma}$ = 0.6502 eV) is energetically more favorable in comparison to the $\pi$ mode ($E_{\rm ads}^{\pi}$ = 0.1545 eV) 14. A similar effect is observed for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on a Au(100) surface where theoretical values of $E_{\rm ads}^{di-\sigma}$ = 0.7790 eV and $E_{\rm ads}^{\pi}$ = 0.2456 eV have been reported 14. It is known that the type of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ bonding depends upon the surface structure. It has been found that the $\pi$-bonded mode dominates in the adsorption of low coordinated atoms and step sites 62. In the case of small metal clusters, almost all atoms are essentially surface atoms that are undercoordinated as compared to the bulk.63 Thus, the $\pi$-bonded configuration of the adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ dominates for small gold clusters. As the number of low coordinated atoms decreases with increase in cluster size, we suppose that the di-$\sigma$ mode will become dominant for large gold clusters.

6 and 7 present the $n$-evolution of the the C–C bond distance, $r_{C-C}$, and the total charge transfer to the adsorbate, $\Delta q^{{\rm C}_{2}{\rm H}_{4}}$, obtained from the natural bond orbitals (NBO) population analysis64. 6 shows that the C–C bond in the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is enlarged up to 1.37–1.39 Å in comparison with the case of a free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (1.32 Å). The $r_{C-C}$ calculated for the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ depends slightly on the cluster size, evolving with $n$ similar to $E_{\rm ads}$. On the other hand, for the di-$\sigma$-bonded ethylene the C–C bond is enlarged up to 1.42–1.50 Å, which is comparable with the length of a single C–C bond in ethane, ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{6}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. The $n$-dependence of $r_{C-C}$ for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ manifests an irregular behavior with maxima at $n=$ 2, 4, 7 and 9. An increase in the C–C bond length, as well as as noticeable change of the bending of the H atoms for di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (see 2), can be explained by the increasingly important role of the $sp^{3}$ hybridization due to the electron donation from the gold cluster to the anti-bonding $\pi^{*}$ orbital of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Therefore, $r_{C-C}$ in the di-$\sigma$-bonded ethylene approaches its typical values for a single C–C bond in $sp^{3}$ hybridized ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{6}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Hence the di-$\sigma$-bonded ethylene is activated strongly in comparison with the $\pi$-bonded one, thus it can be more reactive in catalytic processes.

The total charge transfer to the adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ depends on the balance between the donation and back-donation processes. 7 demonstrates that in the case of the $\pi$ mode of adsorption the total charge transfer $\Delta q^{{\rm C}_{2}{\rm H}_{4}}$ is relatively small and evolves with $n$ similar to $r_{C-C}$. For the di-$\sigma$ mode of adsorption the back-donation from the gold to the $\pi^{*}$ anti-bonding orbital of ethylene dominates over electron transfer from ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ to the cluster. The total charge transfer to ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed in a di-$\sigma$ configuration is large and varies in the range of 0.29 - 0.43 a.u. for $n=$ 2 - 9. It is seen from 7 that $\Delta q^{{\rm C}_{2}{\rm H}_{4}}$ calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ exhibits a pronounced odd-even oscillations as a function of cluster size $n$ with the exception of $n=$1. Note, that for $n=$1 the charge transfer $\Delta q^{{\rm C}_{2}{\rm H}_{4}}$ is small because in this case only one ${{\text{Au}{-}\text{C}}}$ $\sigma$ bond is formed. Similar odd-even oscillations have been found in the $n$-dependence of the adsorption energy of ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters; see, e.g. 56, 65, 23 and references therein. However, the adsorption energy of CO does not present any odd-even effects65.

The appearance of the odd-even oscillations in $E_{\rm ads}$ and $\Delta q^{{\rm C}_{2}{\rm H}_{4}}$ for the di-$\sigma$ mode of adsorption is a result of the electronic shell effects67, 66, 32 in the gold clusters and can be described within the jellium model. In spite of its simplicity, the jellium model can explain, at least on the qualitative level, many physical properties and chemical reactivity of metal clusters.32, 68, 69 Indeed, each Au atom has one $6s$ electron which delocalizes in the whole volume of the cluster. The delocalized electrons are moving in the field of a uniform positive charge background, which binds the valence electron cloud. The gold clusters with an even number of atoms (valence electrons) have closed electronic shell structure. Hence these clusters are more stable and less reactive in comparison with open shell clusters possessing an odd number of atoms (valence electrons).

The prevalence of the electron transfer from the cluster to the adsorbate (back-donation process) is responsible for the promotion of the di-$\sigma$-mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption. Indeed, the di-$\sigma$ mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption on small gold clusters with an odd number of atoms leads to the transfer of unpaired electron from the gold cluster to the ethylene anti-bonding $\pi^{*}$ orbital, causing the weakening of the C–C bond. The neutral gold clusters with even number of valence electrons possess a closed electronic shell structure, hence electron transfer to the adsorbate and formation of the di-$\sigma$ bonded species is suppressed.

8 demonstrates the evolution of molecular adsorption energy, $E_{\rm ads}$ and C–C bond distances, $r_{C-C}$, as a function of cluster size $n$ for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on the anionic {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} (left column) and cationic ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ (right column) clusters with $1\leq n\leq 10$.

The adsorption energy of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on a gold cluster depends on the balance between donation and back-donations processes, hence it can be manipulated by the cluster charge. It is seen form 8a and 8b that excess of the negative or the positive charges on gold clusters strongly influence the molecular adsorption. Thus, the interaction of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ with the negatively charged gold clusters {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} becomes weaker if compared with the neutrals; while excess of the positive charge results in strengthening of the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ bond.

8a demonstrates that the $\pi$-mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption is not stable for {{\text{Au}_{\vphantom{\text{}}\text{}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} and {{\text{Au}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}. The adsorption energy of the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on the gold cluster anions {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} is 0.17 eV in the case of a trimer {{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}, and this increases with an increase in the cluster size: up to 0.82 eV at $n=$7 and dropping to 0.54 eV at $n=$8, while slowly increasing with an increase in the cluster size up to $n=$10.

8b shows that the size dependence of $E_{\rm ads}$ for $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on the gold cluster cations ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ is very different from that of {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}}. Thus, ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ readily adsorbs on the cationic monomer ${{\text{Au}_{\vphantom{\text{}}\text{}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ and dimer ${{\text{Au}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ with an adsorption energy of 2.95 eV and 3.58 eV, respectively. Further increase in cluster size results in the sharp decrease in $E_{\rm ads}$ up to 2.06 eV for $n=$3 followed by its slow decrease for 3$\leq n\leq$6. At $n=$7 the adsorption energy $E_{\rm ads}$ exhibits a maximum followed by the second sharp decrease at $n=$ 8 and 9.

For any $n$ in the range 1$\leq n\leq$10, the adsorption energy of the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on cationic $E_{\rm ads}^{\pi}(Au_{n}^{+})$, neutral, $E_{\rm ads}^{\pi}(Au_{n})$, and anionic, $E_{\rm ads}^{\pi}(Au_{n}^{-})$, gold clusters satisfies the relation: $E_{\rm ads}^{\pi}(Au_{n}^{+})>E_{\rm ads}^{\pi}(Au_{n})>E_{\rm ads}^{\pi}(Au_{n}^{-})$.

Indeed, the excess of the positive charge on the cluster favours the process of electron transfer from the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ to the gold cluster, which stabilizes the $\pi$ mode of adsorption, and strengthens the Au–C bond. A similar effect has been found for adsorption of CO on small anionic and cationic gold clusters71, 70. It was found that the adsorption energy of CO is larger in the case of a cluster of cations and smaller in the case of anions. With the increasing cluster size, the differences in adsorption energies for different charge states become smaller71.

It is important to mention that strengthening the Au–C bond for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on the gold clusters with an excess of the positive charge is responsible for a cooperative effect in simultaneous adsorption of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on neutral gold clusters.43 Indeed, the adsorption of ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ results in an electron transfer from the gold cluster to the anti-bonding $2\pi^{*}$ orbital of ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. In this case, ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ effectively adsorbs on the positively charged gold cluster. A similar effect has also been predicted for adsorption of propene on ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters.40 It was supposed that the binding of propene to ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ should be stronger than the binding to the ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ cluster due to the effective charge transfer from the gold cluster to the oxygen molecule.40

The excess of the positive or the negative charge on gold clusters results in a considerable change in the adsorption energy of the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. As has been discussed above, the adsorption energy of the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on small neutral gold clusters exhibit an odd-even oscillations as a function of the cluster size. These oscillations appear as a result of the electronic shell effects in gold clusters: ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ with odd $n$ have one uncoupled electron that can easily transfer to the empty $\pi^{*}$ anti-bonding orbital of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Such an electron transfer favors the di-$\sigma$ configuration of the adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$.

8a demonstrates that there is no odd-even oscillations in the adsorption energy of the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on the anionic {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} clusters. In that case, the cluster always possesses a weakly bounded electron that it can transfer to the adsorbate.

In the case of the cationic ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters, the electron back-donation process to ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is suppressed for any $n$. 8b shows that $E_{\rm ads}$ calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ exhibit two profound minima at $n=$3 and 9. In accordance with the spherical jellium model, ${{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ and ${{\text{Au}_{\vphantom{\text{}}\text{9}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters possess closed shell electronic structure with 2 and 8 valence electrons, respectively. Hence, ${{\text{Au}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ and ${{\text{Au}_{\vphantom{\text{}}\text{9}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ are chemically inactive, in particular in the processes where electron transfer from the cluster to the adsorbate is responsible for formation of the chemical bond.

8c and 8d demonstrate the $n$-evolution of the C–C bond distance, $r_{C-C}$, in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} and ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,+}}}}$ clusters, respectively. The C–C bond in the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on the anionic and the cationic clusters is enlarged up to 1.36–1.40 Å  irrespective to cluster charge. This is 0.04 - 0.08 Å larger than that for the free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ (1.32 Å). The di-$\sigma$-bonded ethylene is activated strongly in the case of adsorption on the anionic {{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{\,,$--$,}}}} clusters. Thus, in the di-$\sigma$-bonded ethylene the C–C bond is enlarged up to 1.53–1.42 Å for the anionic and only up to 1.48–1.37 Å for the cationic gold clusters. Such an effect has a clear explanation. The excess of electrons on a cluster promotes back-donation of an electron from the cluster to the $\pi^{*}$ anti-bonding orbital of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, hence promoting its activation. The excess of the positive charge on a cluster suppresses the back-donation process, thereby suppressing the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ activation. Thus, by changing the cluster charge one can manipulate the cluster reactivity.

3.3 Vibrational spectra of ethylene adsorbed on small gold clusters

Analysis of the vibrational spectrum of adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ directly reveals the mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption and the state of hybridization. This is due to the sensitivity of the C–C stretching frequency to the hybridized state of carbon atoms.11 In 1985 Stuve and Madix introduced the so-called $\pi\sigma$ parameter for characterizing rehybridization in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ upon adsorption on metal surfaces.11 The $\pi\sigma$ parameter depends on the shift of the C–C stretching frequencies in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ranges from 0 for the free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, to 0.38 for ${{\text{K}\nolinebreak{\text{[}}\text{}\nolinebreak{\text{(}}\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak{\text{)}}\text{}\nolinebreak\text{PtCl}_{\vphantom{\text{}}\text{3}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak{\text{]}}\text{}}}$ (Zeise’s salt), and finally to 1 for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{Br}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Zeise’s salt and ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{Br}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ have been chosen by Stuve and Madix as models describing pure $\pi$- and di-$\sigma$-bonding of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, respectively. Thus the larger the shift in vibrational frequencies (compared to the free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$), the greater the degree of rehybridization.

A similar approach can be applied for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on metal clusters. 1 presents the fundamental frequencies of free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ calculated in the harmonic approximation and obtained from experiment72.

The obtained theoretical results are in a good agreement with experimental data. Some discrepancy, especially noticeable in the high frequency range, might result from anharmonic effects. The anharmonic corrections can be taken into account, for example by the method of n-mode coupling representations of the quadratic force field.73 However such a consideration goes far beyond the scope of the present paper. In order to distinguish between $\pi$ and di-$\sigma$ modes of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption one should to calculate the shift in the C–C stretching frequencies of the adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ with respect to the free molecule. In that case the anharmonic corrections to the vibrational energy levels can be partly canceled. Therefore in this paper, we perform analysis of the vibrational spectra of the adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ in the harmonic approximation.

Following Ref.11, we have selected two vibrational modes: $\nu_{6}$ and $\nu_{8}$, corresponding to the direct C–C and the in-plane ${{\text{CH}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ scissoring modes of vibrations, respectively. Information on the shift of the vibrational frequencies of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ upon its adsorption can be obtained from infrared or Raman spectroscopy. 9 presents the frequency dependence of the Raman scattering activity calculated for a ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ molecule adsorbed in the $\pi$ (9a) and di-$\sigma$ (9b) configurations on the neutral ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters with $n$ = 1, …, 10. The stretching modes corresponding to the $\nu_{6}$ and $\nu_{8}$ vibrations in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ are marked by the vertical arrows. It is seen from 9a that the spectral dependences of the Raman scattering activity calculated for the $\pi$-bonded ethylene on ${{\text{Au}_{\vphantom{\text{}}\text{n}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters are similar for all the cluster sizes $n$ considered. The first maximum at   960 cm*-1* corresponds to the symmetric and asymmetric swinging of H atoms in the direction perpendicular to the plane of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$— similar to the $\nu_{2}$ and $\nu_{3}$ vibrational modes in free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. These vibrations remain unchanged in the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Two intensive maxima at   1280 cm*-1* and   1560 cm*-1* correspond to the $\nu_{6}$ and $\nu_{8}$ vibrational modes in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. These lines are red-shifted in comparison with the free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$.

Solid lines in 10 demonstrate shifts $\Delta\nu_{6}^{\pi}$ and $\Delta\nu_{8}^{\pi}$ in harmonic frequencies $\nu_{6}$ and $\nu_{8}$ due to the $\pi$ mode of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption on small neutral gold clusters. For the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, the scissoring mode $\nu_{8}$ is shifted by 30-40 cm*-1* towards low frequencies if compared with the $\nu_{6}$ mode. Both the $\Delta\nu_{6}^{\pi}$ and $\Delta\nu_{8}^{\pi}$ shifts vary in the range of 58 - 142 cm*-1* and exhibit similar dependence on the cluster size with local minima at $n$=6 and 8. The relatively small frequency shifts $\Delta\nu_{6}^{\pi}$ and $\Delta\nu_{8}^{\pi}$ correspond to the values of the $\pi\sigma$ parameter typical for the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$.

9 demonstrates that the spectral behavior of the Raman scattering activity calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ is different from that obtained for the $\pi$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Thus, for cluster sizes $n$= 2, 4, 5, 7 and 9, a broad band appears in the frequency region of 900 - 1300 cm*-1*. This band results from the strong mixing of the C–C stretching mode $\nu_{6}$ with the hydrogen swinging ($\nu_{2}$ and $\nu_{3}$) and twisting ($\nu_{4}$) modes. The coupling of the $\nu_{2}$, $\nu_{3}$, $\nu_{4}$, and $\nu_{6}$ vibrational modes occurs as a result of considerable deformations in ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. These deformations include the bending of H atoms in respect to the plane of a free ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ due to the $sp^{3}$ hybridization in di-$\sigma$ bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$, as well as the twisting of ${{\text{CH}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ groups in respect to each other along the C–C bond. Such twisting is especially noticeable for ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ clusters. The strong coupling between different vibrational modes makes identification of $\nu_{6}$ and $\nu_{8}$ vibrations to be rather difficult: the mixed modes form new levels, neither of which represent the true $\nu_{6}$ and $\nu_{8}$ states.

Appearance of the $sp^{3}$ hybridization character in di-$\sigma$ bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ results in a considerable red-shift in the stretching frequencies. Such shifts $\Delta\nu_{6}^{{\rm di-}\sigma}$ and $\Delta\nu_{8}^{{\rm di-}\sigma}$ in harmonic frequencies $\nu_{6}$ and $\nu_{8}$ are shown by the dashed lines in 10.

It is seen that the $\Delta\nu_{6}^{{\rm di-}\sigma}$ shift is considerably larger than the $\Delta\nu_{8}^{{\rm di-}\sigma}$ which can be explained by the strong coupling between the $\nu_{2}$, $\nu_{3}$, $\nu_{4}$, and $\nu_{6}$ vibrational levels. The $\Delta\nu_{6}^{{\rm di-}\sigma}$ and $\Delta\nu_{8}^{{\rm di-}\sigma}$ frequency shifts varies in the range of 200 - 370 cm*-1* for $n$=2, … ,10. These values correspond to the $\pi\sigma$ parameter typical for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$. Thus, the analysis of the vibrational modes of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on gold clusters allows one to identify the specific modes of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption and the rate of rehybridization upon adsorption.

4 Conclusions

In summary, we have demonstrated that ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ can be adsorbed on small gold clusters in two different configurations, corresponding to the $\pi$- and di-$\sigma$-bonded species. Adsorption in the $\pi$-bonded mode dominates over the di-$\sigma$ mode over all considered cluster sizes $n$, with the exception of the neutral ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}{-}\text{}\nolinebreak\text{Au}_{\vphantom{\text{}}\text{5}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ system. We found a striking difference in the size dependence of the adsorption energy of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ bonded to neutral gold clusters in the $\pi$ and di-$\sigma$ configurations. The adsorption energy, calculated for the di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ exhibits pronounced odd-even oscillations, showing the importance of the electronic shell effects in the di-$\sigma$ mode of ethylene adsorption on gold clusters. The different rate of hybridization in the $\pi$- and di-$\sigma$-bonded ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ can be responsible for the different rate of catalytic activation and hence, different reactivity of adsorbed ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$.

We have also demonstrated that the interaction of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ with small gold clusters strongly depends on the cluster charge. Hence ethylene adsorption and reactivity can be manipulated by the cluster charge. The strengthening of the ethylene – cluster interaction for gold clusters with an excess of the positive charge is responsible for the cooperative effect in simultaneous adsorption of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ and ${{\text{O}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ on neutral gold clusters. This effect can play an important role in the mechanism of catalytic oxidation of alkenes by dioxygen on the surface of gold clusters.

Finally we have shown that the analysis of the vibrational modes of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorbed on gold clusters allows one to identify the specific modes of ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ adsorption, the rate of rehybridization upon adsorption, and thus the ${{\text{C}_{\vphantom{\text{}}\text{2}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}\nolinebreak\text{H}_{\vphantom{\text{}}\text{4}}^{\vphantom{\text{}}\vphantom{\text{\smash[t]{2+}}}\text{}}}}$ reactivity.

In the present work, we have considered adsorption of ethylene molecule onto small gold clusters consisting of up to 10 atoms. Many interesting problems beyond the scope of the present work arise when considering multiple molecular adsorption on gold nanoparticles of larger sizes up to 1-5 nm, where the strong dependence of the catalytic activity of gold nanoparticles has been observed experimentally. In particular, multiple molecular adsorption on the surface of gold nanoparticles can result in a considerable change in cluster morphology, even for relatively large cluster sizes.

Another important direction for future development is to investigate the effect of alloying on the adsorption and catalytic activation of alkenes on gold clusters. The doping of a gold cluster with atomic impurity can result in considerable changes in the geometrical and electronic structure of the cluster, thereby modifying and controlling the catalytic activity. Understanding how to enhance and control chemical reactions on a cluster surface is therefore a vital task for nanocatalysis.

{acknowledgement}

This work was supported by the Global COE Program (Project No. B01: Catalysis as the Basis for Innovation in Materials Science) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. Calculations were performed in the Research Center for Computational Sciences, Okazaki Research Facilities.